On the occurrence of condensations in steady axisymmetric jets

Abstract

In experiments on the stability of a submerged axisymmetric jet at Reynolds numbers large compared with unity, Reynolds (1962) observed axisymmetric ‘condensations’ which appear to grow spontaneously, whereas Batchelor & Gill (1962) have shown that infinitesimal disturbances of this type do not grow in an inviscid fluid. Here it is shown that axisymmetric disturbances do not grow in a slightly viscous fluid either, and the solution for which the rate of damping is smallest is found. It is suggested that the growth of small but finite disturbances is responsible for the condensations observed. The order of magnitude of the disturbance velocity at which non-linear effects could produce growth of disturbance is found to depend on the wave-number of the disturbance. The smallest velocity which a ‘finite’ disturbance may have is found to be of order $R^{-\frac{2}{3}$, and corresponds to a disturbance whose wave-number is of order $R^{\frac{1}{3}$, R being the Reynolds number based on the local radius and maximum velocity of the jet. On the assumption that some disturbances whose velocity is of this order will grow, deductions are made as to the size, position, wave-number, and point of appearance of condensations. The deductions appear to agree with the experimental results.